A rectangle made of elastic material will be made into a cylinder by joining edges AD and BC. To support the structure, a wire of fixed length L is placed along the diagonal of the rectangle. Find the angle $\displaystyle \theta$ that will result in the cylinder of maximum volume.

see attached picture

thank you very much!

unfortunately i cannot attach a picture
so here is a shot at it...

d._________c
..|............/|
..|........./...|
..|....../......|
..|.../.........|
a|/............|b
theta is angle a, in the triangle cab

2. Hello, doctorgk!

A rectangle made of elastic material will be made into a cylinder by joining edges AD and BC.
To support the structure, a wire of fixed length L is placed along the diagonal of the rectangle.
Find the angle $\displaystyle \theta\;(\angle CAB)$ that will maximize the volume of the cylinder.
Code:
    D * - - - - - - - * C
|             * |
|           *   |
|      L  *     |
|       *       | L·sinθ
|     *         |
|   *           |
| * θ           |
A * - - - - - - - * B
L·cosθ
We have: .$\displaystyle AB \,=\,L\cos\theta,\;CB \,=\,L\sin\theta$

The height of the cylinder is: .$\displaystyle h \:=\:CB \:=\:L\sin\theta\;\;{\color{blue}[1]}$

$\displaystyle AB$ is the circumference of the circular base of the cylinder.
. . $\displaystyle L\cos\theta \,=\,2\pi r\quad\Rightarrow\quad r \:=\:\frac{L\cos\theta}{2\pi}\;\;{\color{blue}[2]}$

The volume of a cylinder is: .$\displaystyle V \;=\;\pi r^2h\;\;{\color{blue}[3]}$

Substitute [1] and [2] into [3]: .$\displaystyle V \;=\;\pi\left(\frac{L\cos\theta}{2\pi}\right)^2(L\ sin\theta)$

. . and we have: . $\displaystyle \boxed{V \;=\;\frac{L^3}{4\pi}\,\sin\theta\cos^2\!\theta}$

And that is the function to be maximized . . . good luck!

3. ## So far

i got it so far, but just a quick question on the optimization. when u differentiate would you factor out the (L^3)/4 pi and then differentiate the trig functions, or would you differentiate the entire thing?

4. Hello,

I think it's better factorising, less obstruction, more visibility

5. ## what do i do next?

so i got the derivative to be...
$\displaystyle \frac{dV}{d\theta}=(\frac{L^3}{4\pi})\cos^3\theta - \sin\theta2\cos\theta\sin\theta$

then i set it equal to zero and i get

$\displaystyle 0=\cos^3\theta - \sin\theta2\cos\theta\sin\theta$

what do i do from here?

6. Originally Posted by doctorgk
so i got the derivative to be...
$\displaystyle \frac{dV}{d\theta}=(\frac{L^3}{4\pi}) {\color{red}(}\cos^3\theta + \sin\theta2\cos\theta\-sin\theta{\color{red})}$

then i set it equal to zero and i get

$\displaystyle 0=\cos^3\theta + \sin\theta2\cos\theta$

what do i do from here?
Hm... No... You have to keep the factor L^3/4pi for the whole thing (correction in red)

The derivative will be :
$\displaystyle \underbrace{\cos^3(\theta)}_{\text{You found it}}+\sin(\theta) 2 (\underbrace{-\sin(\theta)}_{\text{derivative of cos}}) \cos(\theta)$

7. ## then i do

so then the derivative is

$\displaystyle 0=\cos^3\theta - \sin\theta2\cos\theta\sin\theta$

then i can factor out a cosine and get
$\displaystyle 0=\cos^2\theta - 2\sin^2\theta$

i know u have to solve for theta, but i kinda forgot how to solve trig equations

8. You can factor out only if it's $\displaystyle \neq 0$

The better way to do is to factorise, and solve...

9. ## so...

$\displaystyle 0=\cos\theta(\cos^2\theta - 2\sin^2\theta)$

how do u then solve it?
im sure you can factor out the L^3/4 pi because it cannot be zero

10. Here, yes, you can factor it out.

Now, replace $\displaystyle \cos^2(x)$ by $\displaystyle 1-\sin^2(x)$ and then solve for it.

$\displaystyle ab=0 \text{ iff } a=0 \text{ or } b=0$